2
Introduction

2.1 Background

Mathematical models are widely utilized across various scientific and engineering fields to analyze and address real-world issues. However, the accuracy of these models can be compromised due to the inherent simplifications made during their development. To simplify the analysis process, assumptions are often employed, leading to a decrease in model precision. Conversely, as the pursuit of greater accuracy intensifies, the resulting models become more complex, rendering problem-solving equally intricate. Striking the right balance between model simplicity and precision is therefore crucial. In recent years, there have been significant developments aimed at improving the accuracy and efficiency of mathematical models.

These advancements have sparked researchers' interest in exploring new methods and approaches to achieve more robust solutions for applied problems. However, as we delve deeper into the complexities of real-world systems, limitations arise that can hinder progress and even bring it to a standstill. One critical challenge in modeling complex systems is the presence of uncertainties. Real-world scenarios are rarely deterministic, and interval uncertainties play a significant role in affecting their behavior. Interval analysis provides a robust framework for addressing these uncertainties within mathematical models. By expressing variables as intervals rather than single values, interval analysis allows for a more realistic representation of system behavior, considering the potential range of values rather than relying on point estimates.

The relevance of interval analysis in solving problems with uncertainties must be considered. It offers a means to balance oversimplified models that overlook uncertainties and overly complex models that hinder problem-solving. By incorporating interval analysis techniques, researchers and engineers can obtain more accurate and reliable results, accounting for the inherent uncertainties in real-world systems. This book explores the concept of interval analysis and its application in solving problems affected by interval uncertainties. It aims to provide researchers, scientists, and engineers with a comprehensive understanding of interval analysis techniques and their practical implementation.

By employing interval analysis, practitioners can enhance the accuracy and reliability of their models and gain valuable insights into the behavior of complex systems. Through carefully examining interval analysis methods and their applications, this book aims to address the limitations imposed by oversimplification and excessive complexity, ultimately facilitating more effective problem-solving in diverse fields. These limitations include the following:

1. A series of mathematical models include parameters, most of which cannot be accurately calculated in practice. There are many sources of parametric uncertainty, two of the most important of which are [1] as follows:
   • Measurement error is one of the largest sources of indeterminacy in parameter estimation, resulting from measurement instruments and environmental factors.
   • Errors in parameter estimation can occur due to incorrect classification or estimation of parameters with low or unspecified sample sizes. Consequently, this category introduces uncertainty into the model's parameter discussion.
2. Uncertainties in a real model are not the same type; therefore, each one should be appropriately identified within its domain, and its range should be defined. The uncertainties can be divided into three different classes:
   • Epistemic uncertainty: Mostly, there are several reasons for disregarding or lacking sufficient information about the physical system, environment, or estimation of system parameters, etc. [2].
   • Aleatory (random) uncertainty: This uncertainty is caused by random processes that arise from the nature of the actual system or are influenced by the environment, such as noise and similar scenarios.
   • Stochastic uncertainty: Stochastic uncertainty considers the system's high sensitivity to the initial conditions [3].
   • Each model (even without uncertainty) has its specific solution that cannot be applied to another one simultaneously. Random uncertainties are often represented using probability density functions (PDFs), whereas epistemic uncertainties are frequently depicted through fuzzy, interval, and stochastic variables.

For this reason, appropriate methods related to each type of problem should be used to obtain a reliable response from a system considering these uncertainties.

2.2 Relationship Between Error and Uncertainty

In general, the difference between the measured value and the actual value of a system is the calculated error value [4]. Some error values are generated outside the calculation; for instance, there are cases where the inputs are not adequately measured, or even some of the measured information is lost. Additionally, when the system modeling is based on simplification, it may ignore a large part of the system parameters, and these omissions are also considered uncertainties [5].

Other sources of error arise from internal resources based on the discrete nature of digital computing due to constraints such as computational time, storage capacity, or program complexity. Compatibility constraints, such as floating-point representation and conversion, also contribute to these issues.

These problems make it nearly impossible to perfectly align the original system with an approximate model, often involving reducing, rounding, and simplifying.

Precisely measuring the error of a numerical algorithm is typically challenging. In the real world, determining the exact value of the output error for a numerical program is impossible [6]. Consequently, attempts to find an approximate error to solve problems often prove unsuccessful. However, due to the widespread application of approximate engineering approaches, they have become valuable tools in engineering. Over the years, researchers have proposed various methods to handle and address these errors in the presence of uncertainties.

Among the many methods used in this field, Fuzzy methods [7, 8], statistical methods [9], and interval methods [10] are particularly popular.

Each of the mentioned methods has its disadvantages. For example, a Fuzzy method can perform well if we possess complete knowledge of the system or if an expert with extensive information about the scheme can guide the learning process and account for all unknown uncertainties.

Statistical methods are helpful when statistical data, such as mean value and variance, are available. However, this information is not always accessible. On the other hand, the interval methodology employs techniques that only require upper and lower bounds to assess the system, which is typically the case in most modeling scenarios. Figure 2.1 illustrates the variables involved in statistical, Fuzzy, and interval methods.

As depicted in Figure 2.1, the probability variable is determined using the PDF, the Fuzzy variable is based on its membership functions, and the interval variable only requires lower and upper range values. To gain a deeper understanding of the significance of the interval methodology and its distinctions from other methods, we will briefly summarize definitions provided by experts.

Figure 2.1 Representation of the statistical (a), Fuzzy (b), and interval (c) variables.

2.3 Expert Perspectives on Interval Analysis

• The difference between Fuzzy and interval sets is that a Fuzzy set has a membership function that estimates the extent (within the interval [0, 1]) to which any member belongs to the set. Interval sets, on the other hand, do not have such membership functions, and all the members of an interval are indistinguishable because we cannot say that any member holds a higher status in the set than the other.
• This is true: By definition, every algorithm that works for Fuzzy numbers can also be applied to interval numbers. The opposite is also true: thanks to Zadeh's extension principle; to compute, for example, the range Y of a Fuzzy unavailable f(x1, ., xn) over Fuzzy numbers X1, ., Xn, it is sufficient to compute, for each, the range f(x1(a), ., xn(a)) over the crisp a-cuts Xi(a). This way, we obtain an alpha-cut Y(a) for Y. From this perspective, Fuzzy computations reduce to several instances of an interval computation problem. This is precisely how many books on Fuzzy (such as Klir and Yuan) introduce computing with Fuzzy numbers.
• From this viewpoint, it is not true that there are different algorithms for interval and Fuzzy variables. There are some papers in which the authors described algorithms for the general Fuzzy case, clearly understanding (and explicitly writing) that the interval case is a particular case. Some other researchers use the interval algorithms for alpha-cuts, thus getting Fuzzy algorithms.
• Contrary to the impression you seem to have, there are no separate interval and Fuzzy methods. We have two equivalent problems, and any form of solving one problem helps solve the other.
• From the research...